Whakaoti i te 15x^2+4x-4=0

$$15 { x }^{ 2 } +4x-4=0$$

Whakaoti mō x

$$x=-\frac{2}{3}\approx -0.666666667$$

$$x=\frac{2}{5}=0.4$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 15x^{2}+ax+bx-4. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=4 ab=15\left(-4\right)=-60$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōrunga te a+b, he nui ake te uara pū o te tau tōrunga i tō te tōraro. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -60.

$$-1,60 -2,30 -3,20 -4,15 -5,12 -6,10$$

Tātaihia te tapeke mō ia takirua.

$$-1+60=59 -2+30=28 -3+20=17 -4+15=11 -5+12=7 -6+10=4$$

Ko te otinga te takirua ka hoatu i te tapeke 4.

$$a=-6 b=10$$

Tuhia anō te 15x^{2}+4x-4 hei \left(15x^{2}-6x\right)+\left(10x-4\right).

$$\left(15x^{2}-6x\right)+\left(10x-4\right)$$

Tauwehea te 3x i te tuatahi me te 2 i te rōpū tuarua.

$$3x\left(5x-2\right)+2\left(5x-2\right)$$

Whakatauwehea atu te kīanga pātahi 5x-2 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(5x-2\right)\left(3x+2\right)$$

Hei kimi otinga whārite, me whakaoti te 5x-2=0 me te 3x+2=0.

$$x=\frac{2}{5} x=-\frac{2}{3}$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Ko ngā whārite katoa o te āhua ax^{2}+bx+c=0 ka taea te whakaoti mā te whakamahi i te tikanga tātai pūrua: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. E rua ngā otinga ka puta i te tikanga tātai pūrua, ko tētahi ina he tāpiri a ±, ā, ko tētahi ina he tango.

$$15x^{2}+4x-4=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 15 mō a, 4 mō b, me -4 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-4±\sqrt{4^{2}-4\times 15\left(-4\right)}}{2\times 15}$$

Pūrua 4.

$$x=\frac{-4±\sqrt{16-4\times 15\left(-4\right)}}{2\times 15}$$

Whakareatia -4 ki te 15.

$$x=\frac{-4±\sqrt{16-60\left(-4\right)}}{2\times 15}$$

Whakareatia -60 ki te -4.

$$x=\frac{-4±\sqrt{16+240}}{2\times 15}$$

Tāpiri 16 ki te 240.

$$x=\frac{-4±\sqrt{256}}{2\times 15}$$

Tuhia te pūtakerua o te 256.

$$x=\frac{-4±16}{2\times 15}$$

Whakareatia 2 ki te 15.

$$x=\frac{-4±16}{30}$$

Nā, me whakaoti te whārite x=\frac{-4±16}{30} ina he tāpiri te ±. Tāpiri -4 ki te 16.

$$x=\frac{12}{30}$$

Whakahekea te hautanga \frac{12}{30} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 6.

$$x=\frac{2}{5}$$

Nā, me whakaoti te whārite x=\frac{-4±16}{30} ina he tango te ±. Tango 16 mai i -4.

$$x=-\frac{20}{30}$$

Whakahekea te hautanga \frac{-20}{30} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 10.

$$x=-\frac{2}{3}$$

Kua oti te whārite te whakatau.

$$x=\frac{2}{5} x=-\frac{2}{3}$$

Ngā Upane mō te Whakaoti i te Pūrua

Ko ngā whārite pūrua pēnei i tēnei nā ka taea te whakaoti mā te whakaoti i te pūrua. Hei whakaoti i te pūrua, ko te whārite me mātua tuhi ki te āhua x^{2}+bx=c.

$$15x^{2}+4x-4=0$$

Me tāpiri 4 ki ngā taha e rua o te whārite.

$$15x^{2}+4x-4-\left(-4\right)=-\left(-4\right)$$

Mā te tango i te -4 i a ia ake anō ka toe ko te 0.

$$15x^{2}+4x=-\left(-4\right)$$

Tango -4 mai i 0.

$$15x^{2}+4x=4$$

Whakawehea ngā taha e rua ki te 15.

$$\frac{15x^{2}+4x}{15}=\frac{4}{15}$$

Mā te whakawehe ki te 15 ka wetekia te whakareanga ki te 15.

$$x^{2}+\frac{4}{15}x=\frac{4}{15}$$

Whakawehea te \frac{4}{15}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{2}{15}. Nā, tāpiria te pūrua o te \frac{2}{15} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{4}{15}x+\left(\frac{2}{15}\right)^{2}=\frac{4}{15}+\left(\frac{2}{15}\right)^{2}$$

Pūruatia \frac{2}{15} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{4}{15}x+\frac{4}{225}=\frac{4}{15}+\frac{4}{225}$$

Tāpiri \frac{4}{15} ki te \frac{4}{225} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x^{2}+\frac{4}{15}x+\frac{4}{225}=\frac{64}{225}$$

Tauwehea x^{2}+\frac{4}{15}x+\frac{4}{225}. Ko te tikanga pūnoa, ina ko x^{2}+bx+c he pūrua tika pūrua tika pū, ka taea taua mea te tauwehea i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{2}{15}\right)^{2}=\frac{64}{225}$$

Tuhia te pūtakerua o ngā taha e rua o te whārite.

$$\sqrt{\left(x+\frac{2}{15}\right)^{2}}=\sqrt{\frac{64}{225}}$$

Whakarūnātia.

$$x+\frac{2}{15}=\frac{8}{15} x+\frac{2}{15}=-\frac{8}{15}$$

Me tango \frac{2}{15} mai i ngā taha e rua o te whārite.

$$x=\frac{2}{5} x=-\frac{2}{3}$$